Branched coverings of graphs and uniformization theory Alexander - - PowerPoint PPT Presentation

Branched coverings of graphs and uniformization theory

Alexander Mednykh

Sobolev Institute of Mathematics, Novosibirsk State University

Shanghai Jiao Tong University, 12 July 2016

Alexander Mednykh (NSU) 1 / 26

Introduction

The theory of Riemann surfaces was founded in classical works by B. Riemann and A. Hurwitz. In their papers the Riemann surface was defined as a branched covering over the sphere. Starting with 1900 the most important approach to Riemann surface theory was related with uniformization theory created by F. Klein, A. Poincare and P. Koebe. Over the last decade, a few discrete versions of the theory of Riemann surfaces were created.

1 Bacher, R., de la Harpe, P., and Nagnibeda,T., 1997 2 Urakawa, H., 2000 3 Baker, M., Norine, S., 2009 4 Caporaso, L., 2011

In these theories, the role of Riemann surfaces is played by graphs. As well as branched coverings are replaces by quasi-coverings of graphs.

Alexander Mednykh (NSU) 2 / 26

Introduction

Dictionary

1 Riemann surface

⇐ ⇒ Finite connected multigraph

2 Holomorphic map ⇐

⇒ Harmonic map (branched covering) (quasi-covering)

3 The sphere

⇐ ⇒ Tree

4 Torus (= one "hole" surface) ⇐

⇒ Flower (= one cycle graph)

5 Genus (♯ of "holes") ⇐

⇒ Genus (♯ of independent loops)

6 Conformal automorphism ⇐

⇒ Automorphism acting harmonically (= acting free on semi-edges)

Alexander Mednykh (NSU) 3 / 26

Harmonic maps and branched coverings

One of the first definitions of branched covering for graphs was done by T.

• D. Parsons, T. Pisanski and B. Jackson (1980). The main idea was to find

a discrete version of branched covering for graph through dual voltage assignment. Following Baker-Norine (2007) we prefer to give the following geometric definition. For any vertex x of a graph G we denote by Stx the star of G at x.

Definition

A morphism ϕ : G → G ′ is called to be branched covering (also quasi-covering, harmonic map and so on in the literature ) if for all vertices x ∈ V (G), the quantity |ϕ−1(e′) ∩ Stx| is independent of the choice of edge e′ ∈ E(Stϕ(x)).

Alexander Mednykh (NSU) 4 / 26

Riemann-Hurwitz formula for graphs

Recall the classical Riemann-Hurwitz formula. Given surjective holomorphic map ϕ : S → S′ between Riemann surfaces of g and g′, respectively, one has 2g − 2 = deg(ϕ)(2g′ − 2) +

• x∈S

(rϕ(x) − 1), (1) where rϕ(x) denotes the ramification index of ϕ at x. Let G be a finite group of conformal automorphisms acting on S and ϕ : S → S′ = S/G is the canonical map induced by the group action. Then the above formula can be rewritten in the form 2g − 2 = |G|(2g′ − 2) +

• x∈S

(|G x| − 1), (2) where G x stands for the stabiliser of x in G and |G x| = rϕ(x) is the order

• f a stabiliser.

Remark that S has only finite number of points with non-trivial stabiliser.

Alexander Mednykh (NSU) 5 / 26

Riemann-Hurwitz formula for graphs

The latter formula has a natural discrete analogue. By a graph we mean a finite connected multigraph without loops. We define genus of graph X as g = |E(G)| − |V (G)| + 1, that is as cyclomatic number of G. Let G be a finite group acting on graph X without fixed and invertible edges. Denote by g′ genus of the factor graph X ′ = X/G. Then by [Baker-Norine, 2009] we have g − 1 = |G|(g′ − 1) +

• x∈V (X)

(|G x| − 1), (3) where V (X) is the set of vertices of X. We extend the above mentioned results to group actions with fixed and invertible edges.

Alexander Mednykh (NSU) 6 / 26

Finite group action on graphs

We say that a group G acts on X if G is a subgroup of Aut(X). Let X be a finite connected graph. We note the genus g(X) = |E(X)| − |V (X)| + 1 coincides with the Betti number of X that is the rank of the first homology group H1(X, Z). Let G be a finite group acting on the graph X. An edge {x, ¯ x} ∈ E(X) consisting of two semi-edges x and ¯ x is said to be invertible by G if there is an element g ∈ G such that g sends x to ¯ x and ¯ x to x. An edge {x, ¯ x} ∈ E(X) is said to be fixed by G if there is a non-trivial element g ∈ G that fixes x and ¯

• x. We say that G acts on X without

invertible edges if X has no edges invertible by G. Also, G acts on X without fixed edges if X has no edges fixed by G.

Alexander Mednykh (NSU) 7 / 26

Groups acting on a graph without edge reversing

Our first result is the following theorem for groups acting on a graph without edge reversing.

Theorem 1 (M., 2013)

Let X be a graph of genus g and G is a finite group acting on X without invertible edges. Denote by g(X/G) genus of the factor graph X/G. Then g − 1 = |G|(g(X/G) − 1) +

• v∈V (X)

(|G v| − 1) −

• e∈E(X)

(|G e| − 1), where V (X) is the set of vertices, E(X) is the set of edges of X, G x stands for the stabiliser of x ∈ V (X) ∪ E(X) in G and |G x| is the order of a stabiliser.

Alexander Mednykh (NSU) 8 / 26

Groups acting on a graph with invertible edges

Let now G be a finite group acting on a graph X, possibly with invertible

• edges. In this case, there are three different ways to define the factor graph

X/G. 1◦. The factor graph with loops (X/G)loop. 2◦. The factor graph with semi-edges (X/G)tail 3◦. The factor graph without semi-edges (X/G)free.

Alexander Mednykh (NSU) 9 / 26

Groups acting on a graph with invertible edges

Alexander Mednykh (NSU) 10 / 26

Groups acting on a graph with invertible edges

We have the following two theorems.

Theorem 2 (M., 2013)

Let X be a graph of genus g and G is a finite group acting on X, possibly with invertible edges. Denote by g(X/G)loop genus of the factor graph (X/G)loop. Then g − 1 = |G|(g(X/G)loop − 1) +

• v∈V (X)

(|G v| − 1) −

• e∈E(X)

(|G {e}| − 1), where V (X) is the set of vertices, E(X) is the set of edges of X, G v stands for the vertex stabiliser at v ∈ V (X) ∪ E(X) in G, G {e} stands for the stabiliser of the set consisting of two semi-edges of e ∈ E(X) and |G x| is the order of a stabiliser.

Alexander Mednykh (NSU) 11 / 26

Groups acting on a graph with edge reversing

Theorem 3 (M., 2013)

Let X be a graph of genus g and G is a finite group acting on X, possibly with invertible edges. Denote by γ = g(X/G)tail genus of the factor graph (X/G)tail. Then g − 1 = |G|(γ − 1) +

• v∈V (X)

(|G v| − 1) −

• e∈E(X)

(|G e| − 1) +

• e∈E inv(X)

|G e|, where V (X) is the set of vertices, E(X) is the set of edges of X, G x is the stabiliser of x ∈ V (X) ∪ E(X) in G, and E inv(X) is the set of invertibile edges of X.

Alexander Mednykh (NSU) 12 / 26

Harmonic group action on graphs

Let X be a finite connected multigraph without loops.

Definition

A group G < Aut(X) acts harmonically on a graph X if and only if it acts freely on the set of arcs of X. We have the following observation.

Observation

If group G acts harmonically on a graph X then the canonical projection X → X/G is a branched covering.

Alexander Mednykh (NSU) 13 / 26

Harmonic group action on graphs

Let finite group G acts harmonically on a graph X. Then |G e| = 1 for each e ∈ E(X). We have the following corollary from Theorem 3 (compare with Baker-Norine, 2009 and Corry, 2011).

Corollary

Let X be a graph of genus g and G is a finite group acting on X harmonically, possibly with invertible edges. Denote by g(X/G)free genus

• f the factor graph (X/G)free. Then

g − 1 = |G|(g(X/G)free − 1) +

• v∈V (X)

(|G v| − 1) + |E inv(X)|, where V (X) is the set of vertices, E(X) is the set of edges of X, G v is the stabiliser of v ∈ V (X) in G, and E inv(X) is the set of invertible edges of X.

Alexander Mednykh (NSU) 14 / 26

Hurwitz and Accola-Maclachlan theorems

Recall some classical results for Riemann surface theory. For each g ≥ 2 define N(g) := max{|Aut(Sg)| : Sg is a compact Riemann surface of genus g}. Then 8(g + 1) ≤ N(g) ≤ 84(g − 1), and these bounds are sharp in the sense that both the upper and lower bound are attained for infinitely many values of g. The upper bound was found by Hurwitz (1893). The lower bound was independently obtained by

• R. Accola (1968) and C. Maclachlan (1969).

Alexander Mednykh (NSU) 15 / 26

Wiman’s theorem

Klein’s quartic curve, x3y + y3z + z3x = 0, admits the group PSL2(7) as its full group of conformal automorphisms.

• Fig. 1. Klein’s curve with 168=84(3-1) automorphisms.

This is the curve of smallest genus realising the upper bound 84(g − 1) on the order of a group of conformal automorphisms of a curve of genus g > 1, given by A. Hurwitz in 1893. Around the same time, A. Wiman (1895) characterised the curves w2 = z2g+1 − 1 and w2 = z(z2g − 1), g > 1, as the unique curves of genus g admitting cyclic automorphism groups of the largest and the second largest possible order (4g + 2 and 4g, respectively).

Alexander Mednykh (NSU) 16 / 26

Harmonic group action on graphs

Denote by N(g) maximum size of a finite group acting harmonically on a graph of genus g ≥ 2.

Theorem (Scott Corry, 2011)

For g ≥ 2 we have 4(g − 1) ≤ N(g) ≤ 6(g − 1). The upper and lower bound are attained for infinitely many values of g. Recent paper by Scott Corry (2013) states that maximal graph groups G with |G| = 6(g − 1) are exactly the finite quotients of the modular group Γ =< x, y|x2 = y3 = 1 > of size at least 6.

Alexander Mednykh (NSU) 17 / 26

Wiman’s theorem

We obtain the following discrete version of the Wiman theorem.

Theorem 4 (A. Mednykh and I. Mednykh, 2013)

Let X be a graph of genus g ≥ 2 and ZN is a cyclic group acting harmonically on X. Then N ≤ 2g + 2. The upper bound N = 2g + 2 is attained for any even g. In this case, the signature of orbifold X/ZN is (0; 2, g + 1), that is, X/ZN is a tree with two branch points of order 2 and g + 1, respectively.

Alexander Mednykh (NSU) 18 / 26

Wiman’s theorem

Alexander Mednykh (NSU) 19 / 26

Wiman’s theorem

Alexander Mednykh (NSU) 20 / 26

Wiman’s theorem

The second and the third largest cyclic groups are given by

Theorem 5 (A. Mednykh and I. Mednykh, 2013)

Let X be a graph of genus g ≥ 2 and ZN is a cyclic group acting harmonically on X. Let N < 2g + 2 then N ≤ 2g. The upper bound N = 2g is attained only in the following cases: (i) N = 2g and X/ZN is an orbifold of the signature (0; 2, 2g), g ≥ 2; (ii) N = 12 and X/ZN is an orbifold of the signature (0; 3, 4), g = 6. Moreover, the upper bound N = 2g − 1 is attained only in two cases: (iii) N = 3 and X/ZN is an orbifold of the signature (0; 3, 3), g = 2; (iv) N = 15 and X/ZN is an orbifold of the signature (0; 3, 5), g = 8.

Alexander Mednykh (NSU) 21 / 26

Oikawa theorem

In 1956 Kotaro Oikawa proved the following theorem.

Theorem (Oikawa, 1956)

Let Sg be a closed Riemann surface of genus g and A is a finite subset of Sg consisting of |A| ≥ 1 elements. Suppose that 2g − 2 + |A| > 0 and G is a group of conformal automorphisms of Sg leaving the set A invariant. Then |G| ≤ 12(g − 1) + 6|A|. In the next section we find a discrete version of the Oikawa’s. Again, the key point of the proof is the Riemann-Hurwitz relation.

Alexander Mednykh (NSU) 22 / 26

Oikawa’s theorem for graphs

Our result for graphs is the following theorem.

Theorem 6 (R. Nedela, A. Mednykh, 2013)

Let X be a graph of genus g and A is a subset of vertices of X consisting

• f |A| ≥ 1 elements. Suppose that g − 1 + |A| > 0 and G is a finite group

acting on X harmonically and leaving the set A invariant. Then |G| ≤ 2(g − 1) + 2|A|. The upper bound is sharp and is attained for arbitrary large values of g and |A|. So, at least infinitely many often.

Alexander Mednykh (NSU) 23 / 26

Two Arakawa’s theorems

Now our aim is to find discrete versions of two Arakawa’s theorems (2000). The first one states that if G be a finite group of automorphisms of a compact Riemann surface X of genus g ≥ 2 and A and B are two disjoint G-invariant subsets of X of the orders |A| ≥ |B| ≥ 1 then |G| ≤ 8(g − 1) + |A| + 4|B|. The second theorem asserts that if A, B and C are three disjoint the G-invariant subsets of X with |A| ≥ |B| ≥ |C| ≥ 1 then |G| ≤ 2(g − 1) + |A| + |B| + |C|.

Alexander Mednykh (NSU) 24 / 26

Two Arakawa’s theorems

We present a discrete version of the first Arakawa’s theorem by the following theorem.

Theorem 7 (R. Nedela, A. Mednykh and I. Mednykh 2013)

Let X be a graph of genus g ≥ 2 and A and B are two disjoint subsets of vertices of X of the orders |A| ≥ |B| ≥ 1. Suppose that G is a finite group acting harmonically on X and leaving the sets A and B invariant. Then |G| ≤ 3(g − 1) + |A| + 3|B| 2 . Again, the upper bound is sharp and is attained for arbitrary large values of g, |A|, |B| and |C|.

Alexander Mednykh (NSU) 25 / 26

Two Arakawa’s theorems

A discrete version of the second Arakawa’s theorem is given by the following theorem.

Theorem 8 (R. Nedela, A. Mednykh and I. Mednykh, 2013)

Let X be a graph of genus g ≥ 2 and A, B and C are three disjoint subsets

• f vertices of X of the orders |A| ≥ |B| ≥ |C| ≥ 1. Suppose that G is a

finite group acting harmonically on X and leaving the sets A, B and C

• invariant. Then

|G| ≤ g − 1 + |A| + |B| + |C| 2 . As in the two previous theorems, the upper bound is sharp and is attained for arbitrary large values of g, |A|, |B| and |C|.

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